[FOM] Re: The width of V

[Roger Bishop Jones]

On Wednesday 09 July 2003  7:06 pm, Eric Steinhart wrote:

> From: Roger Bishop Jones <rbj at rbjones.com>
>
> >The power set axiom says nothing about what subsets there
> >are, it just says that such subsets as there are, are
> >the members of the power set.
>
> That's quite right, but the question is about the axiom that
> makes V wide; that will be the power set axiom regardless of
> how subsets are defined.  Other axioms may say what subsets
> there are, but the gathering of them together into the next
> successor stage is done by the power set (in standard
> definitions of V).

If you are considering the definition of V(alpha) in
set theory, then the power set constructor does
determine what is in each level.
However, I don't think this is germane to the
original question, which was not about the definition
of V in some suitable metatheory, but about the
effects of the axioms of set theory on the structure
of the models which satisfy those axioms.

Most of the respondents clearly did understand
what I was after and offered axioms which probably
do force V to be wider than it would be with the
normal ZFC axioms. (though there are serious
limits to how much can be achieved since no
first order axiom will prevent there being
countable models, which are very thin!)

Let me try harder to make clear the question which
I had in mind.

First let me say that I was definitely not enquiring
about which of the existing axioms of set theory
makes V wide.  I was enquiring about what additional
axioms have been considered which would make V wider
than it is obliged to be by the axioms of ZFC.
So on that ground alone, the power-set axiom doesn't
qualify.

Secondly let me observe that in our informal intuitive
grasp of what "V" is it feels as if height and width
are independent characteristics.
In this context large cardinal axioms are clearly
intended to give extra height, and its natural to ask
whether there are ways of getting extra width.
As well as being an obvious question to ask in its
own right, at an informal level it seems as if
it should be relevant to some important unsolved
problems, like whether CH "is true", since on the
face of it its truth value is settled pretty low
down in V and knowing more about height *shouldnt*
make any difference.
(though this isn't in fact my motivation for
considering the question)

Roger Jones